The interaction of a line vortex with a collinearly aligned jet is a prototypical configuration for various important applications in aeronautics. The purpose of this study is to analyze the impact of the jet flow on the kinematics and dynamics of a trailing vortex. A particular emphasis is on the effect of a variable relative jet–vortex spacing. To this end, we realized four different jet–vortex configurations in a wind tunnel experiment at a chord-based Reynolds number of 1.7×105 using high-speed stereo particle image velocimetry measurements in five transversal planes located between 2 and 26 chords behind the wing. Stochastic analyses reveal that the jet generally contributes an external excitation to the vortex as a function of the mutual spacing. Compared with the configuration without jets, the vortex amplification increases upon reducing the jet–vortex spacing. Most notably, for all but the closest spacing, the vortex response is qualitatively the same, changing mainly in magnitude. For the closest spacing, however, the dynamics is considerably different, which we suspect to be a consequence of jet entrainment. Proper orthogonal decomposition reveals, for the first time, that the collinearly aligned jet tends to excite a progressively broader range of vortex modes as the jet–vortex spacing is reduced. A close examination of the vortex mean flow seems to preclude linear vortex instabilities, while the vortex characteristics hint toward some form of receptivity mechanism to disturbances being located in the free stream. Our analyses are useful to validate simulation tools on configurations combining simultaneous lift and thrust effects.

The kinematics and dynamics of the jet–vortex interaction constitute fundamental problems in fluid dynamics. In particular, it is the prototypical configuration of the near airplane wake and, as such, has important bearings on the wake evolution and its impact on the environment.1,2 Hence, an understanding of the kinematic and dynamic characteristics is of central interest both for control strategies to efficiently reduce the hazard of wake encounters3 and for the initial formation stages of contrails.4,5 The long persistence of trailing vortices of the order of 10 km (103c assuming a chord length c10m)6 or equivalently 102s (assuming a flight speed of the order of 102ms1)2 makes them the principal hazard for following aircraft.3,7 Efficient control strategies are, thus, in need that accelerate their alleviation. Our focus in this study will be on the effect of the jet on the trailing vortex with a particular emphasis on a variable relative jet–vortex spacing. The importance of the relative spacing for the efficiency of control applications has also been noted by Gursul and Wang4 and Margaris et al.8 Our configuration has some conceptual similarity with studies on the (synthetic-)jet actuation of vortices in a control context.9–13 

Simulations of the jet–vortex interaction mostly relied on idealized initial and boundary conditions, i.e., computations are carried out in a (streamwise) periodic domain, initialized with the superposition of a line vortex (Lamb–Oseen, vortex filament) and a (Gaussian) jet flow.1,5,14–18 Since the near and intermediate wake generally deserved much less attention than the far wake, these initial-boundary conditions are an acceptable approximation.14,16 However, the first phase of the wake evolution, called the jet regime, is spatially developing and poorly represented by such simplified initial-boundary conditions. The jet regime refers to the wake dynamics in approximately the first 10s.1,2 Taking the order-of-magnitude estimates 102ms1 and 10m for airplane cruise speed and chord length, respectively, leads to a downstream range of about 102c. The consequences of spatially, rather than temporally, developing dynamics on the jet regime are highlighted as a future issue by Paoli and Shariff.2 

Due to the associated numerical difficulties, the near wake has been experimentally analyzed. Most of these experiments focused on detailed analyses of the very-near wake up to one,4,9 four,19 and eight chords12 behind the wing. A larger downstream range in the same jet–vortex configuration, as studied here, has been analyzed by Brunet et al.15 and Jacquin et al.20 using laser doppler velocimetry. These two studies emphasized the effect of the vortex on the distribution and entrainment of the jet exhaust. Regarding the reverse interaction, Jacquin et al.20 found that the jets have a small influence on the mean vortex characteristics (mean velocity fields and mean center location). However, the vortex fluctuation dynamics was observed to be enhanced by the presence of the jets. The objective of the present study is a detailed characterization and discussion of the mean and fluctuation fields of the vortex up to 26 chords behind the wing with a particular emphasis on the effect of variable jet positioning. To this end, we conducted high-speed stereo particle image velocimetry (PIV) measurements for the same configuration as Brunet et al.15 and Jacquin et al.20 

The paper is structured as follows. In Sec. II, we present the experimental setting, instrumentation, and the different jet–vortex configurations analyzed. The subsequent analyses of the experimental data describe the kinematics and dynamics of the trailing vortex on the basis of the leading-stochastic moments. After discussing the mean vortex flow in Sec. III, the characterization of the unsteady vortex kinematics and dynamics is detailed in Sec. IV. Our analysis of the vortex-center motion in Sec. IV A is complemented by a discussion of the unsteady vortex dynamics in terms of Reynolds stresses in Sec. IV B and a proper orthogonal decomposition in Sec. IV C. We terminate with a conclusion of the main results in Sec. V.

Experiments were conducted at the ONERA center in Le Fauga-Mauzac in an atmospheric, closed-circuit wind tunnel having a rectangular test section of 1.4m width, 1.8m height at the entrance evolving to 1.85m at the exit, and 5.0m length. The divergence of the upper and lower walls (ceiling-floor) at an angle of 0.58° compensates for the streamwise velocity gradient induced by the development of the boundary layer along the test section walls. The turbulence intensity in the empty wind tunnel is less than 0.5%.

The test configuration probed in this study consists of a rectangular wing with two jet generators placed underneath and is identical to the setting studied in Ref. 20. Schematics of the different views of the setup are shown in Fig. 1 along with photographs of the installation and the relevant details in Fig. 2.

FIG. 1.

Schematic of the experimental installation: (a) side view, (b) top view, and (c) view from the back.

FIG. 1.

Schematic of the experimental installation: (a) side view, (b) top view, and (c) view from the back.

Close modal
FIG. 2.

Photographs showing the experimental installation: (a) global installation showing the suspended wing with the bottom-mounted jets, (b) detail of the wing and the jets, (c) details of the transition devices, and (d) detail of the jet outlet geometry and the internal grid.

FIG. 2.

Photographs showing the experimental installation: (a) global installation showing the suspended wing with the bottom-mounted jets, (b) detail of the wing and the jets, (c) details of the transition devices, and (d) detail of the jet outlet geometry and the internal grid.

Close modal

The vortex is generated by an airfoil with a NACA0012 symmetric profile of chord length c=0.125m and wing span b=0.5m, which is suspended from the wind tunnel ceiling by a NACA0015 profile with a chord length of 0.045m. Stabilizing cables have been tensed to prevent lateral oscillations as can be seen in Fig. 2(a). The wing is placed in the center of the test section, with the leading edge set at a distance of 0.695m (or 5.56c) downstream of the inlet section, corresponding to the end of the nozzle of the wind tunnel. The wing tips are located 0.454m (or 3.63c) from the test section side walls. Boundary-layer transition over the airfoil is triggered by a saw-tooth shaped tape of 6mm width and 0.205mm thickness placed z=6.5mm (5.2% of the chord length) downstream of the leading edge. The wing is set at a constant angle of incidence of α=9°.

Two identical jet generators are mounted on NACA0015 supporting structures with 0.1m chord length from the wind tunnel floor. Their axes are positioned at a height 0.876m above the floor, which yields a distance of 0.142m below the wing trailing edge. The lateral jet position is variable, and three different locations associated with a mutual spacing of djet/b{0.44,0.58,0.80} have been assessed [cf. Figs. 1(a) and 1(c) and 2(b)]. Boundary-layers transition is triggered along the external side of the jet casing and the supporting structures by means of bands of 0.3mm carborundum grains set at an axial distance of 0.011m from the leading edges of the jet casing. The jet-generator exit has a diameter of ϕjet=0.01m and is equipped with an internal perforated plate of 1.5mm hole diameter in order to homogenize the jet flow (see Fig. 2). Note that, unlike Jacquin et al.,20 the jets are not heated. The velocity of the jet is designated as the velocity corresponding to the mass flow rate and is set to Ujet=50.2ms1. It serves as a rough estimate of the flow velocity in the jet core. The jet momentum and wing angle of incidence are realistic for a typical lift and drag balance of an aircraft in cruise condition.

For all experiments, the free-stream velocity was set at U=20ms1, implying a chord-based Reynolds number of Rc:=cU/ν1.7×105. As shown in Fig. 1, the Cartesian coordinate system is attached to the trailing edge in the symmetry plane with the coordinates x, y, and z being positive starboard, vertically upward and in the direction of the mean free-stream velocity U, respectively. For convenience, the spatial coordinates are split into the cross-stream coordinate tuple x=(x,y)M2 in the measurement plane M at a fixed streamwise coordinate z.

Stereo Particle Image Velocimetry (PIV) was used in five transversal planes downstream of the test model to measure the two-dimensional, three-component velocity field u(t,x)=(u(t,x),v(t,x),w(t,x)). The measurement planes are located at the constant streamwise coordinates z{250,500,1545,2500,3280}mm, corresponding approximately to dimensionless z/c{2,4,12,20,26}.

The wind tunnel was seeded with DEHS (di-2-ethylhexyl sebacate) tracer particles of density 0.91gcm3 and mean particle size of the order of 0.5μm (always less than 1μm). The laser sheet is generated by an Nd:YLF pulsed Litron laser system with green visible light of 527nm wavelength and 25mJ per pulse. The PIV images are captured by two V710 Phantom cameras (a maximal image rate of 7.5kHz) equipped with a Nikon objective with 105mm focus and Scheimpflug adapter. The cameras are placed opposite to the laser with respect to the sides walls and at each side of the measurement planes. Forward scattering mode is hence achieved. The calibration of the PIV images is realized using a LaVision0.3×0.3m2 calibration body. The image acquisition is realized with the DaVis 8.2.2 program of LaVision at a resolution of 1280 × 800 pixels with a pixel size of 20×20μm2. In each configuration and measurement plane, ten blocks of N =4096 images are taken at a sampling frequency of fs=3kHz, corresponding to a measurement time T=N/fs1.37s.

The reconstruction of the velocity field u=(u,v,w)=(ui) [i=1,,3 along the coordinates (x, y, z), respectively] uses the Lucas–Kanade method to solve the optical-flow problem in a multi-resolution Gaussian pyramid approach in order to prevent convergence to local extrema. This is implemented in the Folki software of ONERA, which is also used to compute the streamwise vorticity ω=u/yv/x.21 Windows of 31 × 31 pixels are used, which yields a spatial resolution of the PIV data of approximately 7mm.

Four different configurations are analyzed. We call no-jet configuration the reference case of the wing alone without the jet generators. Furthermore, three different jet–vortex configurations with relative spacing djet/b{0.44,0.58,0.80} as shown in Fig. 1 have been assessed. The position djet/b=0.80 corresponds roughly to the position of the rolled-up tip vortex and, thus, deserves a special attention, as we confirm later on.

In order to characterize the unperturbed jet flow, PIV measurements have been conducted without the wing model and seeding the jets only. Additionally, the streamwise veolocity w of the jet flow was measured using hot wires (Dantec 55P11) placed 1mm downstream of the jet-exit plane. To measure the whole jet flow, a totality of 169 hot-wire measurements were taken, whereas a single hot wire was placed at either one of the nodes of a 13 × 13 test matrix. The 196 hot-wire probing locations, equispaced at 0.5mm (i.e., 0.05ϕjet), are shown in Fig. 3.

FIG. 3.

Hot-wire measurements of the mean and turbulence characteristics of the jet flow at the exit nozzle: (a) streamwise component of the mean velocity w¯/Ujet and (b) streamwise component of the turbulence intensity 103×uz/Ujet. The markers indicate the measurement matrix and the solid circle the jet nozzle exit outer edge.

FIG. 3.

Hot-wire measurements of the mean and turbulence characteristics of the jet flow at the exit nozzle: (a) streamwise component of the mean velocity w¯/Ujet and (b) streamwise component of the turbulence intensity 103×uz/Ujet. The markers indicate the measurement matrix and the solid circle the jet nozzle exit outer edge.

Close modal

Figures 3–5 show the mean and turbulence characteristics of the jet flow at the jet nozzle using hot-wire measurements (Figs. 3 and 4) and at z/c=1 using PIV measurements (Fig. 5). The streamwise component of the mean velocity is normalized on Ujet [the mean being defined in Eq. (2)], and the turbulence intensity is defined as

u:=13i=13ui2¯,
(1)

where a prime indicates a fluctuation due to Reynolds decomposition (see Sec. III). In the case of the hot-wire measurements, only the streamwise component uz:=u32¯ of Eq. (1) is available.

FIG. 4.

Hot-wire measurements across the jet at the exit nozzle. Solid (dashed) line shows the streamwise component of the mean velocity along a vertical (horizontal) line across the jet center. Dots (crosses) show the streamwise component of the turbulence intensity along a vertical (horizontal) line across the jet center. The extent of the jet nozzle exit plane is indicated by gray shading.

FIG. 4.

Hot-wire measurements across the jet at the exit nozzle. Solid (dashed) line shows the streamwise component of the mean velocity along a vertical (horizontal) line across the jet center. Dots (crosses) show the streamwise component of the turbulence intensity along a vertical (horizontal) line across the jet center. The extent of the jet nozzle exit plane is indicated by gray shading.

Close modal
FIG. 5.

PIV measurements of the mean and turbulence characteristics of the jet flow at z/c=1: (a) streamwise component of the mean velocity w¯/Ujet and (b) turbulence intensity uz/Ujet.

FIG. 5.

PIV measurements of the mean and turbulence characteristics of the jet flow at z/c=1: (a) streamwise component of the mean velocity w¯/Ujet and (b) turbulence intensity uz/Ujet.

Close modal

The hot-wire measurements at the jet-exit nozzle (Figs. 3 and 4) show that the streamwise mean velocity w¯ is close to Ujet, although the jet is not perfectly axisymmetric yet. In particular, the mean-flow profiles shown in Fig. 4 display a left–right asymmetry. The streamwise turbulence intensity of the jet flow at the exit nozzle of uz/Ujet5×103 is of the same order of magnitude as the free-stream turbulence in the empty wind tunnel.

In order to illustrate the downstream evolution of the jet, Fig. 5 additionally shows the streamwise mean velocity and full turbulence intensity (1) at z/=1 (or equivalently z/ϕjet=12.5) obtained with PIV. We observe that both fields are essentially axisymmetric. The contribution from the transversal components of the mean velocity u¯ and v¯ are about one to two orders of magnitude less than w¯ (not shown). The jet flow spreads laterally while preserving a nearly axisymmetric structure up to z/c7 downstream of the exit nozzle (not shown). The radius of the jet evolves from 0.055ϕjet to 2ϕjet over the chord-length separation between the hot-wire and PIV measurement planes in Figs. 4(a) and 5(a), respectively. The resulting spreading rate of the round jet is about 0.12, in close agreement with the regular value of 0.1.22 The turbulence intensity increases sharply within the same distance, from 0.5% to 12%, with respect to the jet velocity Ujet, as a consequence of the turbulent activity in the jet shear layer.

The jet once released is entrained transversely by the wake vorticity as a function of the mutual jet–vortex spacing. In order to obtain a qualitative idea of this entrainment, we conducted additional PIV measurements (as described above), but by seeding the jet alone and not the rest of the incoming flow. This produces images of the jet exhaust identified by the seeding, while the surrounding is free of particles. Figure 6 shows the transverse distribution of the standard deviation of such images corresponding to the jet exhaust, at z/c{4,20} for the different jet–vortex configurations. The vortex and jet positions in the measurement plane are indicated by a cross and a circle, respectively. From Fig. 6, we observe at once that entrainment is enhanced upon reducing the distance between the jet and the vortex, in agreement with Margaris et al.8 

FIG. 6.

Standard deviation of the spatial distribution of the seeding released at the jet in transverse planes at z/c=4 (left column) and z/c=20 (right column) for the various jet–vortex configurations: (a) and (b) djet/b=0.80, (c) and (d) djet/b=0.58, and (e) and (f) djet/b=0.44. The cross indicates the mean location of the vortex center in the given measurement plane. The straight line indicates the trailing edge of the wing. The circle indicates the outflow section of the jet.

FIG. 6.

Standard deviation of the spatial distribution of the seeding released at the jet in transverse planes at z/c=4 (left column) and z/c=20 (right column) for the various jet–vortex configurations: (a) and (b) djet/b=0.80, (c) and (d) djet/b=0.58, and (e) and (f) djet/b=0.44. The cross indicates the mean location of the vortex center in the given measurement plane. The straight line indicates the trailing edge of the wing. The circle indicates the outflow section of the jet.

Close modal

We see that for all jet–vortex configurations, the roll-up of the vortex sheet issued at the wing trailing edge is centered around a location 5% of the half span inward of the wing tip (marked by a cross). The trailing vortex is then subject to a gradual inward drift to at least 10% of the half span at z/c=20. Note that a detailed discussion of the vortex-center motion follows in Sec. IV. This latter vortex-center position corresponds to the closest jet–vortex spacing of djet/b=0.8, chosen for this reason. The more distant jet positions djet/b{0.44,0.58} are inward this location of the fully rolled-up vortex.

In the jet–vortex configuration with djet/b=0.80, the jet is wound up by the trailing vortex right after its release. At z/c=20, the jet fills the entire domain of the vortex and its near periphery. In the second jet–vortex configuration with djet/b=0.58, spiraling-up of the jet by the vortex is delayed, and the core of the vortex is not populated significantly by the jet at z/c=20. Finally, in the jet–vortex configuration with djet/b=0.44, the jet diffuses before significant entrainment into the vortex becomes possible and the absolute contamination of the core remains weak within the downstream range investigated.

Consider an Eulerian flow field f(x,z,t) which will be either one of the velocity components (u, v, w) or the thereof derived streamwise vorticity ω. Let M be one of the five measurement planes at z=const in which we have measured f(x,z,t),xM over the time interval (0,T). We assume f(x,z,t) to be a stationary random process (in t), while it may be non-homogeneous in z. By the ergodic hypothesis, time and ensemble averages can be identified, and we define the mean as23 

f¯(x,z):=limT1T0Tdtf(x,z,t).
(2)

The Reynolds decomposition then reads f(x,z,t)=f¯(x,z)+f(x,z,t), and we call f the fluctuation.

In order to get meaningful vortex parameters, the measurement data have to be corrected for the effect of vortex displacement, usually attributed to the meandering phenomenon.24–27 A discussion of the meandering motion is postponed to Sec. IV A. This correction is straightforwardly implemented in the case of PIV measurements by means of a coordinate transformation into the reference frame following the vortex motion. That is, for z=const, we re-center the snapshots in the respective instantaneous vortex-center position f(x,z,t)f(xX(z,t),z,t) prior to taking averages.28 Techniques for vortex identification are presented in Ref. 29. Here, the vortex center in the measurement plane at the downstream position z is defined by the vector-valued process

X(z,t):=1ΓMMdxxω(x,z,t),t(0,T),
(3)

in R2.30–32 The quantity ΓM is the circulation, or the zeroth-order vorticity integral, defined around the contour M bounding MR2 about the vortex. It is defined (applying Stokes' theorem) by the spatial vorticity integral30,33

ΓM(z):=Mdxω¯(x,z).
(4)

The circulation is a convenient measure for the strength of the vortex.

With no further information, the variations of X(z,t) appear random, and thus, the convenient way to account for it is to consider the leading stochastic moments, namely, the mean and the standard deviation

X¯(z):=1T0TdtX(z,t)andσXi2(z):=(Xi(z,t)¯X¯i(z))2¯(i=1,2),
(5)

respectively.

Radial profiles are obtained by mapping the Cartesian x, y grid onto a polar mesh r,θ having its origin in the instantaneous vortex center X(z,t). This coordinate transformation automatically corrects for the meandering motion. Assuming approximate homogeneity in the azimuthal coordinate, we can further average over θ(0,2π).

All results shown in this section have been mapped to a polar mesh, averaged over the azimuthal coordinate and measurement time [cf. Eq. (2)]. Eventually, the mean over the ten identically prepared runs is taken. In order to avoid overloading symbols, we denote this average with an overline as in Eq. (2) for convenience.

Very low values of the standard deviation between the ten identically prepared runs of the experiment indicates the repeatability of the experiments.

Due to the symmetries of the vortex flow, we can compute the radial profiles of the mean circulation as Γ(r,z)=2πru¯θ(r,z).31,34

Given the radial profiles, several parameters can be defined to characterize the vortex. We define the vortex-core radius

r1(z):=argmaxr(0,)u¯θ(r,z),
(6)

as the radial coordinate of the maximum of the mean azimuthal velocity. We make use of the convention to denote the quantities evaluated at the core radius with a subscript 1; e.g., u¯θ,1(z):=u¯θ(r1,z) and Γ1(z):=Γ(r1,z).

According to Ref. 35, we define the swirl number

q(z):=1.57|u¯θ,1(z)u¯z(r=0,z)u¯z(r,z)|,
(7)

whereas the mean streamwise velocity difference is relative to the free-stream velocity U.

Before discussing the radial profiles of the mean vortex flow, it is illustrative to obtain a global impression of the whole wake flow. To this end, Fig. 7 shows the fluctuation kinetic energy u2 normalized on the free-stream velocity squared U2 for the no-jet and jet–vortex djet/b=0.80 configurations at z/c=2 and z/c=20, respectively. At the first measurement station, the no-jet wake is globally dominated by the vorticity sheet shed from the wing. The vorticity sheet is much less discernible in the djet/b=0.80 jet–vortex configuration, as the dominating contribution comes from the jet flow, having about an order of magnitude larger fluctuation kinetic energy. In either configuration, we see that only the trailing vortex retains a discernible structure in the fluctuation kinetic energy by z/c=20.

FIG. 7.

Fluctuation kinetic energy u2/U2: (a) no-jet configuration at z/c=2, (b) no-jet configuration at z/c=20, (c) jet–vortex configuration with djet/b=0.80 at z/c=2, and (d) jet–vortex configuration with djet/b=0.80 at z/c=20. The black line illustrates the wing.

FIG. 7.

Fluctuation kinetic energy u2/U2: (a) no-jet configuration at z/c=2, (b) no-jet configuration at z/c=20, (c) jet–vortex configuration with djet/b=0.80 at z/c=2, and (d) jet–vortex configuration with djet/b=0.80 at z/c=20. The black line illustrates the wing.

Close modal

Figure 8 compares the radial profiles of the mean azimuthal and streamwise velocity of the vortex in the no-jet and jet–vortex configuration with djet/b=0.80 at the different measurement stations. We observe the profiles to have qualitatively similar shapes in the two configurations and for all measurement planes. The streamwise mean vortex velocity is characterized by a wake-jet coexistence. That is, the wake-like behavior in the core turns into a jet-like flow in its periphery. A wake-like core flow is commonly observed in experiments.25,36,37 This jet-wake characteristic is represented in the Moore–Saffman vortex31 (further details can be found in  Appendix A). Comparing Figs. 8(a) and 8(b), we see that the coexistence gets much more pronounced in the jet–vortex configuration, likely as a consequence of the roll-up of the jet by the vortex. In all cases, the magnitude of the axial velocity increases downstream. The jet flow seems to have a less structural effect on the azimuthal mean-velocity profiles, but strongly affects their maximum intensity.

FIG. 8.

Downstream evolution of the radial profiles of the streamwise and azimuthal mean velocity: (a) no-jet configuration and (b) jet–vortex configuration with djet/b=0.80 spacing (no measurements were conducted at z/c=26 in this configuration). The profiles are averaged over the ten runs at each measurement station, and the error bars, defined as the standard deviation over the runs, indicate the variability between these runs.

FIG. 8.

Downstream evolution of the radial profiles of the streamwise and azimuthal mean velocity: (a) no-jet configuration and (b) jet–vortex configuration with djet/b=0.80 spacing (no measurements were conducted at z/c=26 in this configuration). The profiles are averaged over the ten runs at each measurement station, and the error bars, defined as the standard deviation over the runs, indicate the variability between these runs.

Close modal

These conclusions also hold true for the intermediate jet–vortex configurations djet/b{0.44,0.58} (not shown), which gradually transition between the two extrema shown in Fig. 8. While placing the jet at djet/b=0.44 has almost no effect on the mean vortex flow over the considered measurement range, the intermediate spacing djet/b=0.58 causes deviations from the no-jet reference mainly at the last measurement stations. Jet proximity promotes tangential velocity reduction, as is found for increased level of turbulence by Ghimire and Bailey.38 The initial transition is likely a smooth function of djet; however, there seems to be a critical value 0.58<djet/b<0.80 beyond which significant differences in the mean flow are observed. Previous studies by Jacquin et al.20 and Jacquin and Garnier39 suggest this discontinuity in the dynamics to be related to the entrainment of the jet in the vortex core already during the roll-up in the djet/b=0.80 configuration, while it is not (or late) entrained in the other configurations. The azimuthal mean velocity asymptotically decays as rn with n0.8±0.02 in all configurations and measurement stations. (For comparison, elliptic loading corresponds to n =0.5.31) Slower than potential-vortex decay is the characteristic of the Moore–Saffman vortex and the value of n in agreement with the findings of Ref. 34. Section A details the comparison of the vortex profiles against the Moore–Saffman and Batchelor models.

These findings are supported by Fig. 9(a), showing the downstream evolution of the maximum mean azimuthal velocity and streamwise mean velocity deficit in the core for all configurations. As stated above, the no-jet and djet/b=0.44 configuration have very little difference over the considered measurement range, and the djet/b=0.58 configuration diverges mainly in the last measurement planes. Significantly different downstream evolution of the maxima is observed, however, in the djet/b=0.80 configuration. We note that the streamwise velocity deficit is of the same magnitude in all cases, and the effect of the vortex-superposed jet flow is mainly in the creation of an additional jet flow in the core periphery [cf. Fig. 8(b)]. A little surprising, we observe a much stronger effect of the jet flow on the maximum of the azimuthal mean velocity which we find to decrease by about a factor two in the djet/b=0.80 configuration over the measurement range. These general trends of decaying vortex mean-velocity maxima are analogous to the experiments conducted in grid turbulence.26,28 We find, however, that the vortex core radius remains at an essentially constant value of r1/c6×102 over the measurement range and for all configurations, consistent with the previous findings.20,25–27

FIG. 9.

Downstream evolution of the vortex parameters of the mean velocity in the different configurations: (a) maximum non-dimensional velocities and (b) swirl number q.

FIG. 9.

Downstream evolution of the vortex parameters of the mean velocity in the different configurations: (a) maximum non-dimensional velocities and (b) swirl number q.

Close modal

The information shown in Fig. 9(a) is readily combined into a single parameter, viz., the swirl number q defined in Eq. (7), that determines the local linear stability of the isolated Batchelor vortex. For values larger than about q2.31, the vortex is inviscidly stable to infinitesimal perturbations.40 Viscous instabilities theoretically exist beyond this threshold for all q41 but are presumably of no importance for the present study.

Figure 9(b) shows the downstream evolution of the mean swirl number for the different configurations. In agreement with the previous conclusions, the principal behavior is qualitatively similar in all cases but the closest djet/b=0.80 spacing. Despite this latter configuration, decreasing maximium azimuthal velocity and increasing streamwise mean-velocity deficit combine to downstream-decreasing swirl numbers. However, over the considered measurement range, the minimally attained values of about q5 remain well above the theoretical instability threshold. This suggests linear stability of the trailing vortex in all configurations. Albeit, particularly in the djet/b=0.80 configuration, the Batchelor vortex is a poor model, as can be anticipated from Fig. 8(b), showing a pronounced jet–wake coexistence in the vortex core. A more pertinent stability analysis should use the Moore–Saffman vortex as a base state. The only such analysis known to the authors, assuming an inviscid fluid, is due to Ref. 42. Indeed, this study shows that the Moore–Saffman vortex gradually loses stability as it deviates from the Batchelor vortex and as long as the core is wake-like. That is, the unstable parameter range increases in 0.44n1 and decreases again for n <0.44. For n =0.8, Feys and Maslowe42 report an instability threshold of q =1.7, which is well below the swirl numbers q5 in the experiment for all configurations [cf. Fig. 9(b)].

Figure 10 compares the radial profiles of the mean circulation (4) of the vortex in the no-jet and jet–vortex configuration with djet/b=0.80 at the different measurement stations. The circulation increases sharply within a distance approximately equal to r22r10.1c in all configurations. This corresponds to the vortical zone associated with the vortex (cf. also Ref. 34) The circulation shows continuous, yet mild, increments further outward this radius. The intermediate jet–vortex configurations (not shown) follow the same transition behavior as discussed in connection with Figs. 8 and 9. In the no-jet configuration, apart from the variation that is observed for z/c{12,26} away from the vortex, currently unexplained, the circulation profiles are practically constant over the entire downstream measurement range. This implies that the vortex roll-up is completed within the first two chords behind the wing, consistent with Refs. 27, 43, and 44.

FIG. 10.

Downstream evolution of the circulation: (a) no-jet configuration and (b) jet–vortex configuration with djet/b=0.80 spacing (no measurements were conducted at z/c=26 in this configuration). The profiles are averaged over the ten runs at each measurement station, and the error bars, defined as the standard deviation over the runs, indicate the variability between these runs.

FIG. 10.

Downstream evolution of the circulation: (a) no-jet configuration and (b) jet–vortex configuration with djet/b=0.80 spacing (no measurements were conducted at z/c=26 in this configuration). The profiles are averaged over the ten runs at each measurement station, and the error bars, defined as the standard deviation over the runs, indicate the variability between these runs.

Close modal

On the other hand, the circulation profiles in the djet/b=0.80 jet–vortex configuration show a distinct downstream evolution. Apart from the circulation at z/c=12, the circulation profiles continuously broaden over a radial range restricted to approximately r/c2,,3r1. The asymptotic circulation Γ(r), however, seems to be almost unaffected and remains practically constant. This finding is in principal agreement with Van Jaarsveld et al.34 

For a Lamb–Oseen vortex, the asymptotic circulation Γ(r) is identical to the initial value Γ0 and, hence, a convenient measure for the vortex strength. Assuming a Lamb–Oseen vortex, Γ0 and the circulation at the core radius Γ1 are related according to Γ0=1.397×Γ1.31Figure 11 shows the downstream evolution of Γ1 for the different configurations. Taking an average value of Γ1/cU0.14, we estimate Γ0/cU1.397×0.140.20. Comparing with Fig. 10 shows that Γ0Γ(r). We, thus, define the circulation Reynolds number RΓ:=Γ0/ν3×104. These values of the circulation are in agreement with the previous studies.20,25,26,34

FIG. 11.

Downstream evolution of the circulation at the core radius Γ1 for the different configurations.

FIG. 11.

Downstream evolution of the circulation at the core radius Γ1 for the different configurations.

Close modal

1. Lateral motion of the mean vortex center

A pair of vortices exhibits a downward motion as a consequence of mutual Biot–Savart induction. We employ here a point vortex model of the vortices to evaluate this trend. Such a model agrees with the fact that the separation of the two counter-rotating trailing vortices, which is of the order of the wing span s × b [cf. Fig. 12(a)] with sπ/4, is large compared to the vortex size r2. A crude estimate yields r2/b2.5×102. Using the point vortex model, the translation velocity and trajectory are given by33 

Y¯̇U=1U2Γ0π2b0.01Y¯(z)c=Y¯0c1U2Γ0π2bzc0.140.01zc,
(8)

where Γ0/cU0.2 denotes the circulation of each one of the trailing (point) vortices (cf. Fig. 10). Figure 12(b) shows the vertical downward translation of the mean vortex-center location for the different configurations. Superposing the theoretical descent according to Eq. (8), assuming a fictitious origin at Y¯0/c=0.14, we find acceptable agreement with the no-jet configuration in the last measurement planes.

FIG. 12.

Downstream evolution of the mean location of the vortex center for the different configurations and comparison with the theoretical downward translation according to Eq. (8).

FIG. 12.

Downstream evolution of the mean location of the vortex center for the different configurations and comparison with the theoretical downward translation according to Eq. (8).

Close modal

If the idealization of point vortices was perfectly true, the conservation of momentum would imply that the spanwise separation between the two point vortices was preserved.32Figure 12(a) shows that in the experiment, the vortices tend to converge monotonously toward the symmetry plane (x =0).

2. Vortex-center fluctuation around the mean

The displacement of the vortex and the effect of the jet are the most interesting features of the experiment. Figure 13 shows the growth of the two components of the vortex-center standard deviation σXi, Eq. (5), for the different configurations as a function of the downstream distance. The fact that σX(z)σY(z) over the measurement range indicates that there is no preferred direction of the transverse motion of the vortex. We observe that the vortex-center motion is a function of the mutual jet–vortex spacing. The jet effect remains low and progressive for djet/b{0.44,0.58}; however, for djet/b=0.80, the amplitude shows a steep initial increment, indicative of a changing mechanism of the jet–vortex interaction. In this last configuration, the amplitude increase is very rapid within the first 45 chord lengths behind the wing and seems to saturate afterward at a level approximately 34% of c.

FIG. 13.

Double-logarithmic representation of the downstream evolution of the meandering amplitude (5) for the different configurations and comparison with the scaling law (9).

FIG. 13.

Double-logarithmic representation of the downstream evolution of the meandering amplitude (5) for the different configurations and comparison with the scaling law (9).

Close modal

Downstream enhancing fluctuation levels of the vortex-center motion is a generally observed phenomenon, usually referred to as meandering. An important feature of the meandering amplitude [defined as the standard deviation Eq. (5)] is its similarity with a diffusion dynamics, which was early on pointed out by Baker et al.24 The idea is that the vortices are essentially displaced as a Lagrangian object by the surrounding free-stream turbulence, which suggests an empirical law for the amplitude growth following

σXi(z)c=2κcUzc(i=1,2),
(9)

where κ denotes an eddy diffusivity.

A comparison of the square-root downstream growth predicted by the empirical meandering-amplitude scaling law Eq. (9) with the different configurations of the experiment is shown in Fig. 13. We find good agreement with σXi(z)/c=103z/c downstream of z/c=4 for the no-jet configuration. For comparison, identification of the eddy viscosity κ with the kinematic viscosity ν would lead to a slope prediction of 2κ/cU2/Rc105/2 with Rc1.7×105 in the experiment.

As expected from the aforementioned conclusions, the empirical law also closely corresponds to the jet–vortex djet/b=0.44 configuration. A significant divergence of the meandering amplitude from this law is, however, observed for the intermediate jet–vortex spacing of djet/b=0.58. Still, the experimental result reasonably follows an amplitude scaling law of the form Eq. (9) if the eddy-viscosity parameter is adjusted appropriately. Unlike these three cases, the meandering amplitude in the djet/b=0.80 configuration is qualitatively different from the empirical law Eq. (9). We note that the vortex response in this last configuration is reminiscent of the findings of Eq. (12), although, in their experiment, the synthetic jet was blowing perpendicular to the trailing vortex at the wing tip. Our findings suggest that the meandering dynamics is actually qualitatively the same in the no-jet and jet–vortex configurations with djet/b{0.44,0.58} (differences are only in terms of the magnitude), while the dynamics is qualitatively and quantitatively different for the closest jet–vortex spacing of djet/b=0.80.

In agreement with Baker et al.,24 this finding strongly suggests that vortex meandering is the consequence of the forcing by disturbances located outside of the vortex. Even more, the dependence on the relative spacing djet indicates the existence of a preferred location for optimal excitation. We recall from the previous study of Ref. 20 that the jet flow does not entrain into the vortex core for djet/b{0.44,0.58} (over the measurement range). This suggests that the effect of the jet on the vortex to be not principally different from that of an enhanced free-stream turbulence intensity, e.g., due to grid turbulence. Indeed, experiments on vortex meandering in grid turbulence support this conclusion.24,34,45

Section IV A indicates how the kinematics of the vortex is dominated by a continuous and increasing lateral motion on the top of a mean downward movement. The dynamics of the vortex is now analyzed through the second-order statistic moments of the velocity field, namely, Reynolds stresses and proper orthogonal decomposition (POD).

Let z/c{4,20} and xM. We define the second central stochastic moment in each space point

Rij(x,z)=ui(x,z)uj(x,z)¯(i,j=1,2,3),
(10)

conveniently referred to as Reynolds stresses. Reynolds stresses did not receive much attention in previous experiments on trailing vortices. In addition to the seminal work by Phillips and Graham,46 Reynolds stresses have been computed by Bailey and Tavoularis,26 Heyes et al.,28 and Beninati and Marshall.36 

A comparison of the Reynolds stresses (10) in the no-jet and jet–vortex configuration with djet/b=0.80 is shown in Figs. 14–17 for the measurement planes located at z/c{4,20}. Analogously to our previous remarks, the intermediate jet–vortex configurations continuously transition between these two extrema.

FIG. 14.

Reynolds stresses at z/c=4 in the no-jet configuration: (a) transversal fluctuation kinetic energy (u2¯+v2¯)/U2, (b) spanwise-vertical uv¯/U2, (c) spanwise-streamwise uw¯/U2, and (d) vertical-streamwise vw¯/U2. The solid circle of radius r1/c=6×102 and black dot indicate the vortex core and mean center, respectively.

FIG. 14.

Reynolds stresses at z/c=4 in the no-jet configuration: (a) transversal fluctuation kinetic energy (u2¯+v2¯)/U2, (b) spanwise-vertical uv¯/U2, (c) spanwise-streamwise uw¯/U2, and (d) vertical-streamwise vw¯/U2. The solid circle of radius r1/c=6×102 and black dot indicate the vortex core and mean center, respectively.

Close modal
FIG. 15.

Reynolds stresses at z/c=4 in the jet–vortex configuration with djet/b=0.80 spacing: (a) transversal fluctuation kinetic energy (u2¯+v2¯)/U2, (b) spanwise-vertical uv¯/U2, (c) spanwise-streamwise uw¯/U2, and (d) vertical-streamwise vw¯/U2. The solid circle of radius r1/c=6×102 and black dot indicate the vortex core and mean center, respectively.

FIG. 15.

Reynolds stresses at z/c=4 in the jet–vortex configuration with djet/b=0.80 spacing: (a) transversal fluctuation kinetic energy (u2¯+v2¯)/U2, (b) spanwise-vertical uv¯/U2, (c) spanwise-streamwise uw¯/U2, and (d) vertical-streamwise vw¯/U2. The solid circle of radius r1/c=6×102 and black dot indicate the vortex core and mean center, respectively.

Close modal
FIG. 16.

Reynolds stresses at z/c=20 in the no-jet configuration: (a) transversal fluctuation kinetic energy (u2¯+v2¯)/U2, (b) spanwise-vertical uv¯/U2, (c) spanwise-streamwise uw¯/U2, and (d) vertical-streamwise vw¯/U2. The solid circle of radius r1/c=6×102 and black dot indicate the vortex core and mean center, respectively.

FIG. 16.

Reynolds stresses at z/c=20 in the no-jet configuration: (a) transversal fluctuation kinetic energy (u2¯+v2¯)/U2, (b) spanwise-vertical uv¯/U2, (c) spanwise-streamwise uw¯/U2, and (d) vertical-streamwise vw¯/U2. The solid circle of radius r1/c=6×102 and black dot indicate the vortex core and mean center, respectively.

Close modal
FIG. 17.

Reynolds stresses at z/c=20 in the jet–vortex configuration with djet/b=0.80 spacing: (a) transversal fluctuation kinetic energy (u2¯+v2¯)/U2, (b) spanwise-vertical uv¯/U2, (c) spanwise-streamwise uw¯/U2, and (d) vertical-streamwise vw¯/U2. The solid circle of radius r1/c=6×102 and black dot indicate the vortex core and the mean center, respectively.

FIG. 17.

Reynolds stresses at z/c=20 in the jet–vortex configuration with djet/b=0.80 spacing: (a) transversal fluctuation kinetic energy (u2¯+v2¯)/U2, (b) spanwise-vertical uv¯/U2, (c) spanwise-streamwise uw¯/U2, and (d) vertical-streamwise vw¯/U2. The solid circle of radius r1/c=6×102 and black dot indicate the vortex core and the mean center, respectively.

Close modal

We find the fluctuation kinetic energy to have mainly contributions from the transversal components R11+R22=(u2¯+v2¯)/U2, which are shown in the respective top-left Figs. 14(a)–17(a). The contribution from the streamwise fluctuation kinetic energy R33=w2¯/U2 is at least about an order of magnitude less in all but the djet/b=0.80 jet–vortex configuration, viz., u2¯v2¯w2¯. This suggests that the energy-carrying vortex modes are associated with transversal rather than longitudinal motion. We further observe from Figs. 14(a)–17(a) that the fluctuation kinetic energy is approximately axisymmetric and essentially confined to the vortex core (indicated by a bold black circle of radius r1). Although this confinement already applies at the first measurement stations, it becomes gradually more pronounced and sharper as we move downstream. Eventually, we observe that the transversal fluctuation kinetic energy R11+R22 in the no-jet configuration amplifies by about almost an order of magnitude over the measurement downstream range, while the streamwise component w2¯ remains at a nearly constant level (not shown). This also holds for the djet/b=0.44 configuration, whereas w2¯ is amplified by the same factor of ten in the djet/b=0.58 configuration (not shown). It remains, however, an order of magnitude less than the transversal fluctuation kinetic energy. Again the downstream evolution in the djet/b=0.80 configuration is qualitatively different in that all fluctuation kinetic energies are of similar order, remaining approximately constant over the measurement range.

As of the relative magnitudes, similar trends hold for the remaining Reynolds stresses. The dominant contribution comes from the transversal stress R12=uv¯/U2 in all the configurations [Figs. 14(b) and 16(b)], with an amplification of about an order of magnitude, except for djet/b=0.80. Again, the two Reynolds stresses involving the streamwise fluctuation velocity, R13=uw¯/U2 [Figs. 14(c) and 16(c)] and R23=vw¯/U2 [Figs. 14(d) and 16(d)], have much less discernible spatial structure and remain at an approximately constant level. We note that the sharp four-lobed spatial pattern at z/c=20 is already present at z/c=4, although less sharp and accompanied by significant contribution in the core periphery.

The Reynolds stresses in the djet/b=0.80 configuration are associated with qualitatively similar spatial patterns. However, the patterns are less confined to the core, having a broader, stronger, and sharper pattern already at z/c=4. It is remarkable that, besides being less confined, the vortex develops the same Reynolds stress patterns by z/c=20 in all configurations.

1. Background

The proper orthogonal decomposition (POD) is the optimal representation of a stochastic function with respect to the L2-norm, i.e., in terms of the variance contained in a truncated representation.47 Note that the following analysis always assumes measurements being taken in one plane z=const., so, for readability, we suppress explicit dependence.

Let the Eulerian fluctuation field, after Reynolds decomposition, tf(x,t)L2(M) be a stationary random process (cf. Sec. III). The proper-orthogonal decomposition theorem47 then guarantees existence of the expansion

f(x,t)=l=1al(t)ϕl(x)t(0,T),xM,
(11)

such that (an asterisk denoting complex conjugation)

akal*¯=λlδklandMdxϕk(x)ϕl*(x)=δklk,l=1,2,,,
(12)

if and only if

Mdxf(x)f*(x)¯ϕ(x)=λϕ(x)xM.
(13)

We note that (11) constitutes an expansion of the random function f into a deterministic basis span{ϕl}L2(M) of the linear manifold f and random expansion coefficients tal(t). The expansion coefficients are defined by the L2(M)-projection al(t):=Mdxϕl(x)f*(t,x).

In practice, we compute the proper orthogonal decomposition using the method of snapshots.47,48 For this purpose, we note that the actual measurement of the continuous random field yields a discrete sample on the spatiotemporal grid of dimension K × N, where K and N are the number of spatial and temporal measurement points in every measurement plane, respectively. As usual, we call N the number of snapshots and the K×N-matrix F:=(f1,f2,,fN),fq=f((q1)Δt)K for q=1,2,,N, the snapshot matrix. The sampling interval Δt is of the order of the correlation time or larger,48 which guarantees approximate independence of subsequent snapshots.23 Under these conditions, the above proper orthogonal decomposition is equivalent to solving the reduced eigenvalue problem

1Np=1N(fq,fp)bp=λbqq=1,2,,N,
(14)

with (·,·) a (weighted) inner product in K approximating the inner product (12) in L2(M). The expansion of the eigenvalues in Eq. (13) results from ϕ=q=1Nbqfq.

2. POD results

The previous analysis of Reynolds stresses reveals the governing contribution to the vortex fluctuation by the transversal components. The transversal dynamics are conveniently combined into a scalar field as the streamwise vorticity. We, therefore, computed the POD of the fluctuation streamwise vorticity ω in order to characterize the main contributions to the variance of the transversal motion. Previous POD analysis of the vortex-dominated flow has been done on the vorticity44,49,50 as well as on the velocity.12,37

A comparison of the leading POD mode in the no-jet and djet/b=0.80 jet–vortex configuration is shown in Figs. 18 and 19 for z/c=2 and z/c=20, respectively. We recognize the well-known dipolar vorticity pattern confined to the vortex core (indicated by a bold black circle) associated with a displacement wave. The second POD mode is structurally identical but mutually rotated by 90°, while the higher-order modes constitute multi-polar patterns of continuously increasing complexity (not shown). These results are in agreement with the previous experiments, e.g., Refs. 44, 49, and 50. The leading POD modes of the intermediate jet–vortex configurations transition continuously between the patterns shown in Figs. 18 and 19.

FIG. 18.

Leading POD mode of the streamwise fluctuation vorticity in the first measurement plane at z/c=2: (a) no-jet configuration and (b) jet–vortex configuration with djet/b=0.80. Blue (red) color shading and solid (dashed) contours represent positive (negative) values with an arbitrary and not identical scaling. The solid circle of radius r1/c=6×102 and black dot indicate the vortex core and the mean center, respectively.

FIG. 18.

Leading POD mode of the streamwise fluctuation vorticity in the first measurement plane at z/c=2: (a) no-jet configuration and (b) jet–vortex configuration with djet/b=0.80. Blue (red) color shading and solid (dashed) contours represent positive (negative) values with an arbitrary and not identical scaling. The solid circle of radius r1/c=6×102 and black dot indicate the vortex core and the mean center, respectively.

Close modal
FIG. 19.

Leading POD mode of the streamwise fluctuation vorticity in the first measurement plane at z/c=20: (a) no-jet configuration and (b) jet–vortex configuration with djet/b=0.80. Blue (red) color shading and solid (dashed) contours represent positive (negative) values with an arbitrary and not identical scaling. The solid circle of radius r1/c=6×102 and black dot indicate the vortex core and the mean center, respectively.

FIG. 19.

Leading POD mode of the streamwise fluctuation vorticity in the first measurement plane at z/c=20: (a) no-jet configuration and (b) jet–vortex configuration with djet/b=0.80. Blue (red) color shading and solid (dashed) contours represent positive (negative) values with an arbitrary and not identical scaling. The solid circle of radius r1/c=6×102 and black dot indicate the vortex core and the mean center, respectively.

Close modal

At the first measurement station z/c=2, additional to the typical dipolar vortex response mode, the leading POD mode in the djet/b=0.80 jet–vortex configuration contains an important mark of the jet flow outside the vortex core, which disappears by z/c=4. From z/c=4 on, the leading dipolar POD modes in all configurations remain essentially structurally unchanged, while contributions in the free stream gradually disappear. Recalling the definition of the vortex center (3), the greater lateral extent of the leading POD modes upon increasing djet (the jets closer to the wing tips) explains the observed larger lateral motion amplitudes in Fig. 13.

The downstream evolution of the absolute POD eigenvalue (normalized on c2/U2) associated with the leading POD mode is compared for the different configurations in Fig. 20(a). This result is reminiscent of the analysis in Ref. 12.

FIG. 20.

Downstream evolution of the leading eigenvalue of the POD of the fluctuation vorticity for the different jet–vortex interaction configurations: (a) absolute value normalized on c2U02 as in Fig. 21 and (b) relative contribution of the modal variance with respect to the total variance level in the respective measurement planes. Small markers are the results for the individual runs, while the respective bold markers (connected by solid lines) are the mean over all runs.

FIG. 20.

Downstream evolution of the leading eigenvalue of the POD of the fluctuation vorticity for the different jet–vortex interaction configurations: (a) absolute value normalized on c2U02 as in Fig. 21 and (b) relative contribution of the modal variance with respect to the total variance level in the respective measurement planes. Small markers are the results for the individual runs, while the respective bold markers (connected by solid lines) are the mean over all runs.

Close modal

It is instructive to compare the evolution of the absolute eigenvalue [in Fig. 20(a)] with that of the relative share the leading eigenvalue has in the total variance, i.e., λ1/j=1Nλj. This is shown in Fig. 20(b). We see that in this case, the downstream evolution in the different configurations approximately collapse. Keeping in mind that the absolute values increase with djet [cf. Fig. 13(a)], this suggests that the jet tends to excite a wider range of vortex response modes and that this range becomes larger, the smaller the relative jet–vortex spacing is. We support this conclusion by considering the POD eigenvalue spectra (of the first ten eigenvalues) for the different configurations in Fig. 21. While we see the almost exclusive excitation of the leading dipolar POD-mode pair in the no-jet configuration [Fig. 21(a)], progressively more POD modes are amplified downstream as we increase the jet separation distance djet.

FIG. 21.

Eigenvalue (mean over all runs) spectra of the largest ten values from POD of the vorticity covariance for the different configurations and measurement stations: (a) no-jet configuration, (b) jet placement djet/b=0.28, (c) jet placement djet/b=0.21, and (d) jet placement djet/b=0.10. The different measurement stations [see legend in (a)] are shown with the same marker style and color in each configuration.

FIG. 21.

Eigenvalue (mean over all runs) spectra of the largest ten values from POD of the vorticity covariance for the different configurations and measurement stations: (a) no-jet configuration, (b) jet placement djet/b=0.28, (c) jet placement djet/b=0.21, and (d) jet placement djet/b=0.10. The different measurement stations [see legend in (a)] are shown with the same marker style and color in each configuration.

Close modal

We presented a wind tunnel experiment on the jet–vortex interaction for variable relative spacing djet using PIV measurements and a generic apparatus mimicking cruise flight, made of an inclined rectangular wing equipped with two jets underneath. The strength of the jet flow was designed to be realistic for the given setting and was kept the same throughout the different configurations. Despite completing the previous experimental studies (e.g., Ref. 20), the present analysis, providing all the inflow conditions, in particular that for the jet exhaust, constitutes a complete and unique data set to calibrate simulation tools.

Our focus was on the vortex-response characteristics as a function of the different jet positions djet/b{0.44,0.58,and0.80}. The wake flow comprises the trailing vortices that form near the wing tips, the rolling-up vortex sheet issued at the wing trailing edge, and the jets for all but the no–jet configurations. The vortices are found to roll-up within the first two chords behind the wing and to descend under the mutual induction. Their circulation is little influenced by the jet presence and reaches approximately the value 0.2cU. The vortex mean velocity profiles exhibit a close to Gaussian evolution for the azimuthal velocity, while the streamwise mean velocity yields a velocity deficit in the core and a local jet-like overshoot in the core periphery, where the jets are present and entrained, all the more so as the jets are initially located closer to the tips. We show that the no-jet configuration agrees reasonably well with the Batchelor model, while the jet–vortex configurations follow the Moore–Saffman model with an n-parameter roughly equal to 0.8.

The most interesting and largely novel result of the experiment is the persistent and strong unsteady lateral motion of the vortex that is found for all configurations, with increased amplitude as a function of the jets' presence and proximity to the wing tips. The statistical analyses of the leading-order stochastic moments revealed several key conclusions, listed as follows: (i) the motion of the vortices shows no preferred direction within the downstream range of the measurement; (ii) the motion amplitude follows a diffusion-like law in the downstream direction that suggests a similarity of our observations with many other experiments and what is referred to as vortex meandering. Finally, (iii) the closest jet–vortex configuration, with djet/b=0.8, departs from this general behavior by seemingly disrupting the vortices right after their formation close to the trailing edge, and setting a sudden large amplitude of motion. This discontinuous and qualitatively different vortex response upon reducing the mutual jet–vortex spacing beyond a certain point has been reported for the first time.

All governing parameters other than the free-stream turbulence intensity remaining unchanged points toward the net disturbing influence of the jets as a function of djet. This vortex-response characteristic suggests that a preferred external excitation location rc exists. Taking the inward shift of the vortex into consideration, this location can be estimated to be 6<rc/r112. In particular, this result suggests that vortex meandering is neither due to a linear vortex instability exclusively present in the core, nor the consequence of a confining effect of the facility walls. Rather, we have evidence that meandering is, in fact, the vortex response to disturbances of a generic nature situated in the near periphery of the core. It is noteworthy that this conclusion is consistent with the results of recent receptivity studies by Blanco-Rodríguez et al.51 and Bölle et al.52 The pertinence of these and related studies (e.g., Ref. 53) to explain the experimental vortex dynamics is emphasized by the evidence for a linear vortex dynamics implied by our results. Although not conclusive, these conclusions contribute important novel aspects on the question of the origin and nature of unsteady vortex dynamics.

The vortex tends to concentrate practically all the fluctuation kinetic energy in the core which is amplified downstream in all but the djet/b=0.80 configuration, and the main contribution to the Reynolds stresses comes from the transversal components. Based on these findings, we computed the POD of the streamwise fluctuation vorticity. Analysis of the variance downstream evolution in the different configurations shows that the main effect of the jet, besides increasing the absolute vortex-response variance levels, is in the excitation of a progressively larger number of POD modes. To the best of our knowledge, this finding of a broadband excitation of a multitude of vortex response modes by the jet as a function of spacing is original.

We resume that the jets have only a minor effect on the leading-order statistics of the vortex, i.e., the mean flow and the mean drift path of the vortex center, while they do significantly affect the unsteady vortex dynamics, and this influence is a function of the mutual jet–vortex spacing.

This work is based on an experiment that was carried out within the PHYWAKE research project under the funding of the French Ministry for Civil Aviation (DGAC) contract CONVENTION 2015/03.

The authors have no conflicts to disclose.

Tobias Bölle: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Validation (equal); Writing – original draft (equal). Vincent Brion: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Validation (equal); Writing – original draft (equal). Marie Couliou: Conceptualization (supporting); Formal analysis (supporting); Methodology (supporting); Writing – original draft (supporting); Writing – review & editing (equal). Pascal Molton: Data curation (lead); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The radial profile of azimuthal and axial mean velocities is fitted to a selection of vortex models that are often used for this purpose.44Figure 22 shows the comparison of the vortex profile in the no-jet configuration with the Moore–Saffman and Batchelor models, respectively. The Moore–Saffman model offers the best match for the azimuthal mean velocity, while the axial mean velocity is well represented by both models. The n factor of the Moore–Saffman model, representing the decay rate of the azimuthal velocity in the potential flow region, equals approximately 0.8, rather than 1 as expected for the Batchelor vortex and explains the better fitting of the Moore–Saffman model. The same results are found for the jet–vortex configuration djet/b=0.80 (not shown).

FIG. 22.

Comparison of the vortex profile against the Moore–Saffman model for (a) z/c=1 and (b) z/c=5 and the Batchelor vortex for (c) z/c=1 and (d) z/c=5, in the no-jet configuration.

FIG. 22.

Comparison of the vortex profile against the Moore–Saffman model for (a) z/c=1 and (b) z/c=5 and the Batchelor vortex for (c) z/c=1 and (d) z/c=5, in the no-jet configuration.

Close modal

As detailed in Sec. II and shown in Figs. 1 and 2, the jets are installed on bottom-mounted support structures, which a priori must be expected to contribute an additional disturbance to the vortex. In order to estimate the effect of the supports on the vortex, we have conducted additional PIV measurements of the three different configurations djet/b{0.44,0.58,and0.80} with the jets installed but not blowing. On comparing these results against the same configurations but with the jet switched on and the no-jet case, the disturbing effect of the supports can be estimated.

The distinctly different impact exerted by the supports on the vortex is well illustrated by comparing the leading-order vortex-center stochastic moments (discussed in Sec. IV) for the seven different configurations, which we call for convenience no-jet, djet/b{0.44,0.58,and0.80} jet-on, and jet-off configurations. Figure 23, displaying the mean drift trajectories of the vortex center, shows that the variations in the vortex-center mean location are mainly induced by the supports rather than the jet flow.

FIG. 23.

Downstream evolution of the mean location of the vortex center for the different configurations, comparing jet-on and jet-off.

FIG. 23.

Downstream evolution of the mean location of the vortex center for the different configurations, comparing jet-on and jet-off.

Close modal

However, we find that the fluctuation dynamics, being the main interest of this study, is only insignificantly influenced by the support structures. This is shown in Fig. 24, comparing the standard deviation of the vortex-center location σ(z)=σX2(z)+σY2(z) for the various configurations. Particularly for the djet/b{0.44and0.58} jet–vortex configurations, the supports have no effect on the vortex fluctuations at all, as the respective curves in Fig. 24 collapse.

FIG. 24.

Downstream evolution of the standard deviation σ(z)=σX2(z)+σY2(z) of the vortex center for the different configurations, comparing jet-on and jet-off.

FIG. 24.

Downstream evolution of the standard deviation σ(z)=σX2(z)+σY2(z) of the vortex center for the different configurations, comparing jet-on and jet-off.

Close modal
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